Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evalute the following integrals:
$\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}$
✓

Answer

Let $\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}\ .....(\text{i})$
Let $\log\tan\frac{\text{x}}{2}=\text{t}$ then,
$\text{d}\Big[\log\tan\frac{\text{x}}{2}\Big]=\text{dt}$
$\Rightarrow\text{cosec x dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{\text{cosec x}}$
Putting $\log\tan\frac{\text{x}}{2}=\text{t}$ and $\text{dx}=\frac{\text{dt}}{\text{cosec x}}$ in equation (i), we get
$\text{I}=\int\frac{\text{cosec x}}{\text{t}}\times\frac{\text{dt}}{\text{cosec x}}$
$=\int\frac{\text{dt}}{\text{t}}$
$=\log|\text{t}|+\text{C}$
$=\log\Big|\log\tan\frac{\text{x}}{2}\Big|+\text{C}$
$\therefore\text{I}=\log\Big|\log\tan\frac{\text{x}}{2}\Big|+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{\text{x}}\sin{\text{x}}\text{ on }[0,\pi]$
→
Find an equation for the set all points that are equidistant from the planes 3x - 4y + 12z = 6 and 4x + 3z = 7
→
Find the coordinates of the point where the line $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{4}=\frac{\text{z}-2}{2}$ intersect the plane x - y + z - 5 = 0. Also, find the angle between the line and the plane.
→
If x = a (cos t + t sin t) and y = a (sin t – t cos t), 0 < t < $\frac{\pi}{2},$ 

$\text{find }\frac{\text{d}^{2}\text{x}}{\text{dt}^{2}},\frac{\text{d}^{2}\text{y}}{\text{dt}^{2}}\text{ and }\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.

→
Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{4}}_{0}\frac{\sin^2\text{x}\cos^2\text{x}}{(\sin^3\text{x}\cos^3\text{x})}\text{ dx}$
→
Evaluate the following integrals:

$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$

→
Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
→
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{1}{1+\cot\text{x}}\text{ dx}$
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\cos2{\text{x}}\text{ on }[0,\pi]$
→